Problem: You have found the following ages (in years) of 6 turtles. The turtles are randomly selected from the 28 turtles at your local zoo: $ 4,\enspace 50,\enspace 63,\enspace 2,\enspace 82,\enspace 99$ Based on your sample, what is the average age of the turtles? What is the variance? You may round your answers to the nearest tenth.
Because we only have data for a small sample of the 28 turtles, we are only able to estimate the population mean and variance by finding the sample mean $({\overline{x}})$ and sample variance $({s^2})$ To find the sample mean , add up the values of all $6$ samples and divide by $6$ $ {\overline{x}} = \dfrac{\sum\limits_{i=1}^{{n}} x_i}{{n}} = \dfrac{\sum\limits_{i=1}^{{6}} x_i}{{6$ To compensate for this underestimation, rather than simply averaging the squared deviations from the mean , we total them and divide by $n - 1$ $ {s^2} = \dfrac{\sum\limits_{i=1}^{{n}} (x_i - {\overline{x}})^2}{{n - 1}} $ $ {s^2} = \dfrac{{2116} + {0} + {169} + {2304} + {1024} + {2401}} {{6 - 1}} $ $ {s^2} = \dfrac{{8014}}{{5}} = {1602.8\text{ years}^2} $ We can estimate that the average turtle at the zoo is 50 years old. There is a variance of 1602.8 years $^2$.